Notice Bauer Bodoni Bold & User's Guide Manuals

Dynamical ionization ignition of clusters in intense and short laser pulses
D. Bauer∗
Max-Born-Institut, Max-Born-Strasse 2a, 12489 Berlin, Germany

A. Macchi
Dipartimento di Fisica “Enrico Fermi”, Universit`
a di Pisa & INFM, sezione A, Via Buonarroti 2, 56127 Pisa, Italy
(Dated: June 26, 2003)
The electron dynamics of rare gas clusters in laser fields is investigated quantum mechanically
by means of time-dependent density functional theory. The mechanism of early inner and outer
ionization is revealed. The formation of an electron wave packet inside the cluster shortly after
the first removal of a small amount of electron density is observed. By collisions with the cluster
boundary the wave packet oscillation is driven into resonance with the laser field, hence leading
to higher absorption of laser energy. Inner ionization is increased because the electric field of the
bouncing electron wave packet adds up constructively to the laser field. The fastest electrons in the
wave packet escape from the cluster as a whole so that outer ionization is increased as well.
PACS numbers: 36.40.-c, 42.50.Hz, 33.80.-b, 31.15.-p

I.

INTRODUCTION

Clusters bridge the gap between bulk material and single atoms. When placed into a laser field all atoms inside
(not too big) clusters experience the same laser field, contrary to the bulk where a rapidly created surface plasma
and the skin effect prevents the laser from penetrating
deeper into the target. In rarefied gases, on the other
hand, the laser can propagate but the density is too low
to yield high absorption of laser energy and high abundances of fast particles. Hence, the absorption of laser energy is expected to be optimal in clusters. In fact, highly
energetic electrons [1], ions [2–4], photons [5–7], and neutrons originating from nuclear fusion [8] were observed in
laser cluster interaction experiments. A prerequisite for
clusters as an efficient source of energetic particles and
photons is the generation of high charge states inside the
cluster. Several mechanisms for the increased ionization
(as compared to single atoms in the same laser field) have
been proposed. Boyer et al. [9] suggested that collective
electron motion (CEM) inside the cluster is responsible
for inner shell vacancies the corresponding radiation of
which was observed in experiments [10]. Rose-Petruck
et al. [11] introduced the so-called “ionization ignition”
(IONIG) where the combined field of the laser and the
ions inside the cluster leads to increased inner ionization,

i.e., electrons are more easily removed from their parent
ion as if there was the laser field alone. The removal
of electrons from the cluster as a whole is called outer
ionization. An “outer-ionized” cluster will expand because of the Coulomb repulsion of the ions while a heated,
quasi-neutral cluster expands owing to the electron pressure. According to experiments the latter viewpoint of
a nanoplasma (NP) [2, 12] seems to be appropriate for
big clusters of about 104 atoms or more [13] while nu-

∗ E-mail:

bauer@mbi-berlin.de

merical simulations indicate that the Coulomb explosion
dynamics prevail for smaller clusters [14, 15]. For recent
reviews on the interaction of strong laser light with rare
gas clusters see [16, 17].
In this paper we are aiming at clarifying the early ionization dynamics in laser cluster interaction. Because
the plasma has to be created first the NP model is not
applicable. Although we shall find CEM to be indeed
an important ingredient for the early ionization dynamics of clusters, the origin of the CEM should be explained

rather than assumed beforehand. The IONIG model predicts that “as the ionization proceeds beyond the first
ionization stage, strong electric fields build up within the
cluster that further enhance ionization” [11]. Here we are
interested in how these strong inner fields are generated.
Once the cluster is charged after the first electrons are
removed, the remaining bound electrons experience the
attraction of neighboring ions. This attraction will be
largest for bound electrons of ions sitting near the cluster boundary. The other ions will pull these electrons
into the cluster interior. This force might be supported
by the electric field of the laser, thus leading to further
ionization. The IONIG model was successfully called on
for interpreting the experimental results in [18].
Although this IONIG mechanism is appealing, the details of the ionization dynamics remain unclear. If the
force exerted by the other ions is strong enough to ionize
further it should be also strong enough to keep the already freed electrons inside the cluster. Hence, it remains
to be explained how the electrons are removed from the
cluster as a whole (outer ionization). The electrons, after inner ionization, may as well shield the space charge
of the ions so that the IONIG mechanism would come
to an end, and the NP model would take over even in
small clusters. So, why does a strong electric field build

up inside the cluster whose interplay with the electric
field of the laser is constructive for inner as well as outer
ionization?
Classical particle methods are frequently applied to in-

2
vestigate the electron and ion dynamics of clusters in
laser fields [11, 14, 19–23] since a full three-dimensional
quantum treatment is out of reach with nowadays computers. In these classical accounts inner ionization was
either accounted for by sampling the quantum mechanical electron distribution of the bound state by a microcanonical, classical ensemble of electrons, or by using ionization rates so that the electron dynamics was simulated
only after inner ionization. Semi-classical Thomas-Fermi
theory was employed in [15] to study the explosion dynamics of clusters consisting of up to 55 atoms. In our
work we use time-dependent density functional theory
[24–26] since we are mainly interested in the early ionization dynamics of clusters were quantum mechanics is
important.
The article is organized as follows. In Section II the
numerical model is introduced. In Section III results are
presented concerning the groundstate properties of the
model cluster (III A), the electron dynamics (III B), the
formation of collective electron motion inside the cluster

and outer ionization (III C), the effect of the collective
electron motion on inner ionization (III D), the frequency
dependence of outer ionization (III E), and the influence
of the ionic motion (III F). Finally, we conclude in Section IV.
Atomic units (a.u.) are used throughout the paper
unless noted otherwise.

II.

NUMERICAL MODEL

Time-dependent density functional theory (TDDFT) is
employed to study the ionization dynamics of small and
medium size rare gas clusters in intense and short laser
pulses. To that end the spin degenerate time-dependent
Kohn-Sham equation (TDKSE) is solved. However, solving the TDKSE in three spatial dimensions (3D) for
rare gas clusters in laser fields is too demanding for
todays computers. One reason for this is that, contrary to metal clusters [27] or fullerenes [28], the electrons in the ground state of rare gas clusters are not
delocalized so that jellium models where the ionic background is smeared out and assumed spherical are not
applicable. In order to make a numerical TDKS treatment feasible two simplifications were made. First, as

in previous studies of clusters in laser fields [29–32],
the dimensionality of the problem was restricted to 1D,
namely to the direction of the linearly polarized laser
field described by the vector potential A(t) = A(t)ex
in dipole approximation [33]. To that end the Coulomb
interactions were replaced by soft-core Coulomb interactions, i.e., |r − r ′ |−1 → [(x − x′ )2 + a2ee ]−1/2 and
|r − Rk |−1 → [(x − Xk )2 + a2ei ]−1/2 for the electronelectron interaction and the electron-ion interaction, respectively. The smoothing parameters aee and aei may
be chosen to yield ionization energies similar to real 3D
systems. The second simplification was the use of the
exchange-only local density approximation (XLDA) so

that Vxc [n(x, t)] = VXLDA [n(x, t)] = −α(3n(x, t)/π)1/3
where n(x, t) is the electron density. The pre-factor α
would be unity in full 3D XLDA calculations. In the 1D
model it may be chosen to yield satisfactory ground state
properties (we used α = 3/4). The ionic potential was
PNion
Z[(x − Xk )2 + a2ei ]−1/2 with constant
Vion (x) = − k=1
nearest-neighbor distances Xk+1 − Xk = d ≈ 2rWS where

rWS is the Wigner-Seitz radius. One may look at the 1D
ion chain as representing those Nion ions of a 3D spherical cluster which are situated along the diameter parallel
to the linearly polarized laser field. The cluster radius
then is R ≈ (Nion − 1)d/2, and the number of ions in the
3
real 3D cluster N3D = R3 /rWS
would be ≈ (Nion − 1)3 .
The TDKSE for the NKS Kohn-Sham (KS) orbitals
Ψj (x, t), j = 1, . . . , NKS reads
∂
Ψj (x, t) = HKS Ψj (x, t)
∂t

(1)

1
HKS = − ∇2 + U + VXLDA + Vion + Vlaser .
2

(2)

i
where

From the doubly spin-degenerate KS orbitals the total
PNKS
|Ψj (x, t)|2 is calcuelectron density n(x, t) = 2 j=1
R ′
lated. U (x, t) = dx n(x′ , t)[(x−x′ )2 +a2ee ]−1/2 is the 1D
Hartree potential (accounting for the electron-electron
repulsion), and Vlaser = −iA(t)∂x governs the interaction
with the laser (taken in velocity gauge with the purely
time-dependent term ∼ A2 transformed away). Eq. (1)
was solved using the Crank-Nicolson method with the accuracy of the spatial derivatives boosted to fourth order.
Since the Hamiltonian (2) itself depends (through the
density) on the KS orbitals a predictor-corrector method
should be used in combination with the usual CrankNicolson time-propagation. In practice, however, the
evaluation of the Hamiltonian (2) using the density from
the previous time step is usually accurate enough.
For the ion-ion interaction a soft-core potential

Z 2 [(Xj − Xk )2 + a2ii ]−1/2 was assumed as well. The ions
j = 1, . . . , Nion were moved according to their classical,
non-relativistic equations of motion,
M X¨j = Z 2

Nion

X

k=1

Xj − X k
[(Xj − Xk )2 + a2ii ]3/2

−Z

Z

−Z

∂
A(t),
∂t

dx

(3)

n(x, t)(Xj − x)
[(Xj − x)2 + a2ei ]3/2
(4)

using the Verlet algorithm.
The results which will be presented in the Sections
III A–III E were obtained for an ion mass M = 131 · 1836
(Xe atom) and aii = 1. Due to the short laser pulse
durations (< 40 fs) and the modest charge states < 4
created, the ionic motion did not appreciably affect the
ionization dynamics of our 1D model. However, since in a

real 3D cluster consisting of (Nion −1)3 ions the Coulomb
explosion would be more violent than in our 1D model
with Nion ions (of the same ion mass and charge state)
the issue of ionic motion will be discussed separately in
Section III F.
Finally, before presenting our results, we briefly relate
our approach to the work in Refs. [29, 31, 32]. Instead
of the simple XLDA, V´eniard et al. [31] used in their
1D TDDFT approach the more advanced exchange potential proposed by Krieger, Li, and Iafrate (KLI) [34]
but restricted their studies to ion chains up to 7 fixed
atoms with one active shell only. Besides the extensive
study of the harmonic radiation emitted by the 1D cluster, V´eniard et al. focussed on the dependence of ionization on the internuclear distance and on the number
of ions in the chain. This is different from our goal to
highlight the dynamics of IONIG. The methodological
approach instead is very similar. The only differences are
the mobile ions and the choice of simple XLDA instead
of the KLI exchange potential. The latter simplification
gave us the freedom to simulate bigger clusters and more
active shells per atom.
The 1D models in [32] and [29] were called on to study
the explosion dynamics of Xe clusters. Since for this
purpose a full TDDFT treatment with many KS orbitals
is too demanding even in 1D, further simplifications had
to be adopted. In [32] a single orbital, representing all
electrons, was introduced while in [29] time-dependent
Thomas-Fermi theory was used. Our main emphasis in
the present paper is complementary to this work since we
are not dealing so much with the Coulomb explosion of
the cluster but are predominantly interested in the early
electron dynamics.

III.

RESULTS AND DISCUSSION
A.

Groundstate properties

Let us consider a chain of Nion = 9 ions with nearestneighbor distance d = 8 and nuclear charge per ion
Z = 4. Hence, the ground state of the neutral cluster
consists of N = 36 electrons. The smoothing parameters
for the soft-core Coulomb interaction were simply chosen
to be aee = aei = aii = 1. In Fig. 1 the ground state
density and the various contributions to the total potential are plotted: the ion potential Vion , the Hartree
potential U , and the exchange potential VXLDA . Although the ionic potential alone has its absolute minimum at the central ion the total potential, including the
classical Hartree-repulsion and XLDA, consists of nine
almost identical and equidistantly separated potential
wells, each locally similar to that of the corresponding
individual atom. Consequently, the ground state density
displays nine almost equal, well localized peaks, and the
energy levels of the cluster are approximately at the same
positions as those of the atom, namely around −1.17 for
the 2Nion = 18 inner electrons, and around −0.23 for

Potential (a.u.), density (a.u.)

3

x (a.u.)

FIG. 1: Ground state electron density (orange, dasheddotted) and the various contributions to the total potential
(red, solid): ionic potential Vion (black, solid), Hartree potential U (blue, dotted), and XLDA potential VXLDA (green,
dashed).

the 18 outer electrons (note, that in the 1D model there
are only 2 electrons per shell). Since in XLDA Koopman’s theorem is usually not well fulfilled the energy of
the highest occupied orbital does not equal the ionization energy for removing the outermost electron. Calculating the ionization energy from the difference of the
total energy of the neutral and the singly ionized cluster
∆Ecluster = 0.26 was obtained. Doing the same for the
single atom led to ∆Eatom = 0.49 which is a reasonable
value for rare gas atoms.
B.

Electron dynamics

Let us start by comparing the electron motion in the
cluster with that of a single atom. In Fig. 2 contour plots
of the logarithmic density are shown vs. space and time.
The results for the full cluster (a), the single atom (b),
and an artificial cluster made of noninteracting atoms (c)
in a laser pulse of frequency ω = 0.057 (λ = 800 nm) and
rather modest field amplitude Eˆ = 0.033, corresponding
to ≈ 3.9 × 1013 Wcm−2 , are shown. The laser field was
ramped up linearly over 3 cycles, held 8 cycles constant,
and ramped down again over 3 cycles (hereafter called
a (3, 8, 3)-pulse) so that the pulse duration was ≈ 37 fs.
The density in the contour plot (c) was calculated by
assuming that all of the Nion = 9 atoms behave as the
single atom in plot (b).
It is seen that the electron dynamics of the cluster
(a) and the noninteracting atoms (c) differ significantly
from each other already at t ≈ 150 because more electron
density leaves to the right at that time instant in (a) than
it does in (c). This behavior of stronger ionization of the
cluster than of the individual atoms continues during the
subsequent half laser cycles.
A qualitatively different electron dynamics inside the
cluster emerges for t ≥ 400. While in the cluster (a)
an accumulation of electron density bounces from one

4

(b)

(c)

Time (a.u.)

(a)

Space (a.u.)

FIG. 2: Logarithmic density log n(x, t) vs. space and time for
(a) the cluster with Nion = 9, Z = 4 (b) the single atom,
and (c) a cluster made of noninteracting atoms, all with an
electron dynamics as shown in (b). The laser parameters were
ˆ = 0.033, (3, 8, 3)-pulse.
ω = 0.057, E

boundary of the cluster to the other such an electron
dynamics, of course, cannot build up in the ensemble
of independent atoms (c). The formation of an electron
wave packet which travels through the entire cluster in
step with the laser field is remarkable in view of the fact
that the excursion of a free electron in the laser field
ˆ = 0.033 amounts to x
E
ˆ = 10.2 only while the diameter
of the cluster is 2R ≈ 64. By varying the cluster size
(results for Nion = 17 will be presented in the following
subsection) it was found that the formation of the bouncˆ 2 < R/2
ing wave packet at laser intensities where E/ω
is a robust phenomenon, not sensitive to the cluster parameters. Test runs with more advanced exchange potentials such as the Slater potential with self-interaction
corrected XLDA and the KLI potential [34] were performed to ensure that the coherent electron motion is
not an artifact of plain XLDA. However, the wave packet
formation is sensitive to the laser frequency, as will be
shown in subsection III E.

C.

Formation of collective electron motion inside
the cluster and outer ionization

It is useful to study the phase relation between the oscillating electron density inside the cluster and the laser
field in order to understand the formation of the electron wave packet with unexpected large excursion amplitude. Given a laser field E(t) of optical (or lower) frequency, the polarization of an atom is ∼ −E(t) because
the bound electrons are able to follow adiabatically the
force exerted by the field. Hence, the phase lag of the polarization with respect to the laser field is ∆φ = π. Free
electrons, on the other hand, oscillate perfectly in phase
∼ E(t) so that ∆φ = 0. Energy absorption from the laser

R
˙
field is low in both cases because dt x(t)
· E(t) ≈ 0
(with x(t) the expectation value for the position of an
electron). During ionization there is necessarily a transition from ∆φ = π to 0 where energy absorption can take
place. During the ionization of atoms this transition occurs rapidly while in clusters, after inner ionization, the
electrons may be still bound with respect to the cluster
as a whole. Hence, the free motion of electrons ∼ E(t)
comes to an end at latest when they arrive at the cluster
boundary. There, they either escape from the cluster,
contributing to outer ionization, or they are reflected so
that their phase relation with the driving laser is affected,
leading on average to an enhanced absorption of laser
energy. Although collisions of the electrons with ions
are included in our TDDFT treatment the effect of the
boundary on the electron dynamics clearly dominates.
The unimportance of electron ion collisions in medium
size and small clusters was pointed out in [14] while the
relevance of boundary effects was recently affirmed in [35]
within the NP model [36].
In Fig. 2a it is seen that some electrons enter about
ten atomic units into the vacuum before they are pulled
back by the cluster charge. This is reminiscent of what
in laser plasma physics is called Brunel effect [38], “vacuum heating” [39], or, more expressively, “interface phase
mixing” [40]. Thanks to the fact that the fast electrons
leave the cluster (and slow electrons are accelerated) a filter effect comes into play so that a wave packet can form
that oscillates with the laser frequency and an excursion
amplitude of about the cluster radius R.
In order to underpin this scenario the dipole of
the electron density inside the cluster xinner (t) =
R R+d/2
dx xn(x, t) was calculated for several laser and
−(R+d/2)
cluster parameters. The results are shown in Fig. 3. In
panel (a) the laser and cluster parameters were the same
as in Fig. 2. One sees that during the first few laser periods the electrons indeed move ∼ −E(t) (green, dotted
curve) as indicated by the first, green bar at the top of
the xinner -plot. Then, the electrons inside the cluster get
out of phase with the laser for about nine cycles (red bar)
so that ∆φ ≈ π/2. During this period the dipole amplitude x
ˆinner is particularly high [41], and the wave packet
bouncing inside the cluster is clearly visible in Fig. 2. Finally, towards the end of the laser pulse the phase relation
of the few electrons which were removed from their parent ions but did not make it to leave the cluster becomes
that of free electrons (blue bar), i.e., xinner (t) ∼ E(t). In
the lower plot the number of electrons inside the cluster
Ninner is plotted and compared with Nion times the result
for the single atom. It is seen that during the first phase
(green bar) ionization of the cluster proceeds similar to
the single atom case. However, when the phase lag is
shifted to ∆φ = π/2 the cluster continues to ionize while
the single atom ionization comes to an end.
In Fig. 3b the same is shown for a higher laser intensity. Essentially, the phase ∆φ behaves in the same way
but this time, due to the stronger laser field, ionization
of the outer shell is almost completed during the first few

5
(b)

N inner

x inner

(a)

atom
atom

cluster
cluster

(d)

N inner

x inner

(c)

atom
cluster

Time (a.u.)

cluster
atom

Time (a.u.)

FIG. 3: Dipole xinner and number of electrons inside the cluster Ninner vs. time for different laser and cluster parameters.
ˆ = 0.033, ω = 0.057, Nion = 9, Z = 4 (as in Fig. 2);
(a) E
ˆ = 0.114, ω = 0.057, Nion = 9, Z = 4; (c) E
ˆ = 0.114,
(b) E
ˆ
ω = 0.057, Nion = 17, Z = 2; (d) E = 0.099, ω = 0.18,
Nion = 9, Z = 4. The course of −E(t) (a (3, 8, 3)-pulse) is included in the xinner -plots (dotted in green) for distinguishing
motion in phase with −E(t) (green bar at the top edge of the
panel) and motion approximately π/2 and π out of phase (red
and blue bar, respectively). For comparison, Nion times the
result for the single atoms are included in the Ninner -plots.

laser cycles. Hence, the final average charge state is almost the same for the cluster and the single atom. The
period where the phase lag is about ∆φ ≈ π/2 lasts only
a few laser cycles (red bar) and so does the wave packet
motion inside the cluster. In Fig. 3c the result for a bigger cluster (Nion = 17, Z = 2) is presented, revealing a
qualitatively similar scenario as in (a) with the bouncing wave packet surviving for about 9 cycles. Finally, in
panel (d) a higher laser frequency was used (ω = 0.18)
while keeping the cluster parameters as in (a) and (b).
The Ninner-plot reveals that the single atom ionizes more
efficiently than the cluster for these laser parameters. We
will come back to the frequency dependence of outer ionization in subsection III E.

D.

Effect of the collective electron motion on inner
ionization: dynamical ionization ignition

The formation of collective electron dynamics as exposed in the previous subsection explains how the absorption of laser energy is increased due to a phase shift
into resonance with the driving field, and how the electrons, after inner ionization, are efficiently transported
out of the cluster (outer ionization). The increased inner ionization still remains to be analyzed. IONIG states
that the presence of the other ions is responsible for the
more efficient removal of bound electrons. This is because
two neighboring ions form a potential barrier (cf. the Vion
curve in Fig. 1) through which an electron may tunnel
when the whole cluster is submitted to an electric field
so that the entire potential is tilted. However, we found
from our numerical studies that this energetic advantage
of a bound electron inside the cluster as compared with
an electron in the corresponding single ion is not very
pronounced for medium size and small clusters at moderate charge states. Instead, we propose a dynamical
version of IONIG where the previously introduced collective electron motion plays an important role. Coherent
electron motion was suggested to be responsible for inner
shell vacancies in Xe clusters, leading to x-ray emission
[5, 10]. In our numerical model there are only two shells
and we did not find particularly high line emission from
the cluster as compared to the single atom. However, it
is possible that, owing to the lack of dynamical correlation in XLDA, the interaction of the electrons in the
wave packet with the still bound electrons is underestimated in our model. Thus, dynamical IONIG might be
the mechanism behind the experimental results reported
in [5].
Despite the fact that in our mean field approach there
are no “hard” collisions of the electrons in the wave
packet with the still bound electrons, the electric field
associated with the oscillating electron packet already
enhances inner ionization. In Fig. 4 two snapshots of
the total effective potential and the electron density are
presented for the same laser parameters as in Fig. 2. In
panel (a), electron density and total potential are shown
for a time where the electron wave packet is close to the
left cluster boundary (the red bar at the bottom of the
density plot indicates xinner (t)). The contour plot (b)
shows for each ion at position Xi the “difference density”
R Xi +d/2
dx n(x, t) − Ninner/Nion which indicates whether
Xi −d/2
there is a lack of electron density at that position inside
the cluster (black and dark colors) or whether there is
excess density (yellow and light colors) compared to the
average density Ninner/Nion .
In panel (a), at time t = 662.5 the wave packet is close
to the left boundary of the cluster while there is a lack
of electron density near the right boundary. This charge
distribution leads to a force Fint on the other electrons
(pointing to the right) and therefore increases the ionization probability. The electric field of the laser instead
is close to zero so that Flaser is small. The situation ap-

6

Potential (a.u.)

Fint

Number of removed electrons

Flaser

(b)

Time (a.u.)

El. density (a.u.)

(a)

− +

Potential (a.u.)

Flaser

+

−

+

Space (a.u.)

single atom

FIG. 5: Z = 4 minus the electron density, integrated ±d/2
around the central ion in the cluster to determine the number
of removed electrons as a function of time (solid curve, red).
The result for the single atom is also shown (dotted, black).

Time (a.u.)

El. density (a.u.)

Fint

cluster

Time (a.u.)

(d)

(c)

wave packet sweeps
over central ion

Space (a.u.)

FIG. 4: Snapshots of the electron density and the total potential (plus the laser electric field potential alone) at time
(a) t = 662.5 and (c) t = 697.5. The bar at the bottom of
the density plot indicates xinner(t). Arrows, +, and − in the
potential plots illustrate the forces Flaser , Fint exerted by the
laser field and the space charge, respectively. The contour
plots (b) and (c) show whether excess (light colors) or lack
(dark colors) of electron density (with respect to the average
density Ninner /Nion ) prevails at the position of an ion. The
black horizontal lines indicate the times where the snapshots
were taken.

proximately a quarter of a laser cycle later is shown in
panel (c). The wave packet is at the center of the cluster,
moving with maximum velocity to the right and repelling
bound or slow electrons in front of it. The force Flaser is
close to its maximum value at that time, pointing into
the direction in which the wave packet moves. The ionization probability is, again, greater than with the laser
field alone. Thus, during the course of a laser cycle the
total force Flaser + Fint clearly leads to higher ionization
as if there was the laser field only.
The fact that the electron wave packet dynamics indeed increases inner ionization is underpinned by Fig. 5
where the number of electrons in the region ±d/2 around
the central ion in the cluster was subtracted from the initial value Z = 4 and is compared with the corresponding
single atom result. The laser parameters were the same

as in Figs. 2 and 4. While for the single atom the ionization is completed for t > 700 the average electron density
around the central ion in the cluster is still decreasing.
When the electron wave packet sweeps over the central
ion the density is temporarily increased, leading to local
minima in the curve of Fig. 5. When the wave packet
is closest to one of the two cluster boundaries the lack
of electrons around the central ion is maximal. The absolute increase of this maximum each half cycle means
ongoing inner ionization. In contrast, the single atom,
where only the laser field is present but no wave packet
can form, does not ionize any further.
E.

Dependence of outer ionization on the laser
frequency

The interaction of the model cluster with Nion = 9 was
investigated for the two different laser frequencies ωl =
0.057 and ωh = 0.18 (corresponding to 800 and 254 nm,
respectively) and laser intensities between 4 × 1012 and
1016 Wcm−2 . The pulses were of (3, 8, 3)-shape for both
frequencies, that is, the pulse durations were Tl ≈ 37 fs
and Th = 12 fs, respectively. After the laser pulse, the
average charge state in the cluster Zav = Z − Ninner/Nion
was calculated. Note that Zav only yields information
about outer ionization for it does not distinguish between
electrons that are still bound to their parent ions and
those which move inside the cluster.
In Fig. 6 the average charge state is plotted vs. the laser
intensity for the two frequencies ωh and ωl . The results
for the single atom are also shown. As discussed in the
previous subsections, it is seen that in the low frequency
case the atoms in the cluster are stronger ionized than
an individual atom in the same laser field. Both charge
states come close only for Zav = 2, that is when the

7

(b)

(c)

Time (a.u.)

Average charge state

(a)

cluster (lf)
atom (hf)
cluster (hf)

atom (lf)

Laser intensity (W/cm 2)

Space (a.u.)

FIG. 6: Average charge state vs. laser intensity of the cluster
(solid lines) and the individual atom (dotted) for the two different frequencies ωl = 0.057 (lf, drawn red) and ωh = 0.18
(hf, drawn blue). See text for discussion.

FIG. 7: Same as in Fig. 2 but for the higher frequency ωh =
ˆ = 0.099. Logarithmic density log n(x, t) vs. space
0.18 and E
and time for (a) the cluster, (b) the single atom, and (c) a
cluster made of noninteracting atoms.

two electrons of the first shell are removed but the two
electrons of the next shell are still strongly bound. The
stepwise increase of the charge state due to the electronic
shell structure is very pronounced in the low-frequency
cluster case as well as for the single atoms at both, low
and high frequency. The cluster in the high-frequency
field instead shows a very different behavior: between
1014 Wcm−2 and the threshold to the inner shell at ≈
5 × 1015 Wcm−2 , the charge state of the single atom is
higher than the average charge state in the cluster.
In Fig. 7 the dynamics of the cluster electrons is shown
for a (3, 8, 3)-pulse of frequency ωh = 0.18 and peak field
ˆ = 0.099, corresponding to an intensity of
amplitude E
14
3.44 × 10 Wcm−2 . It is seen from Fig. 6 that for this
intensity the single atom ionizes more efficiently than the
cluster as a whole, contrary to what happens at the lower
frequency ωl = 0.057. Fig. 7 reveals that the electrons,
although removed from their parent ions, mostly remain
inside the cluster. From plot (c) one infers that if the
atoms inside the cluster were independent there would be
a strong electron emission for 100 < t < 350. The emitted electrons have sufficient high kinetic energy to escape
from their parent ion (and, thus, from the “independent
atom”-cluster). Contour plot (a), instead, shows that in
the real cluster a significant fraction of the electrons near
the cluster boundaries return due to the space charge
created by all the ions. A wave packet dynamics that
could enhance outer ionization, as in the low frequency
result of Fig. 2, does not form. We attribute this to the
fact that the initial inner ionization occurs less adiabatic
(multiphoton instead of tunneling ionization). Moreover,
the excursion x
ˆ = 3.06 < d/2 is too small to trigger
any collective motion. The dynamics inside the cluster is
rather “splash-like,” as can be inferred from the strongly
fluctuating electron density between the ions in Fig. 7a.
Hence, although at high laser frequencies inner ionization

is high, outer ionization remains low since there is not
the coherent electron dynamics supporting outer ionization. Consequently, for creating quasi-neutral nanoplasmas and suppressing Coulomb explosion the use of high
frequency lasers is favorable. Reduced ionization of clusters in laser fields of, however, many times higher frequency (to be generated by x-ray free electron lasers in
the near future) was also found in the numerical simulations of [21]. On the other hand, in the soft-x-ray FEL
experiment performed by Wabnitz et al. [42] increased
ionization of Xe clusters (as compared to single atoms)
was observed at 98 nm wavelength, 100 fs pulse duration,
and intensities up to 7 × 1013 Wcm−2 . The reason for
this unexpected behavior at short wavelengths is not yet
clear.
For the high frequency ωh and laser intensities I >
3 × 1015 Wcm−2 the average charge state in the cluster
overtakes the charge state of the single atom (see Fig. 6).
It was therefore interesting to check whether under those
conditions a wave packet also forms at the higher frequency ωh . This is indeed the case. However, due to the
rapid ionization the wave packet dynamics lasts only a
few laser cycles (or less).

F.

Influence of ionic motion on the wave packet
formation

So far the ion mass was set to M = 131 · 1836 (Xe
atom). This high mass lead to no appreciable ionic motion in the 1D cluster during the pulse. Even in the
“worst case” where all active electrons are removed at
t = 0 so that the ionic charge Z = 4 is not screened at
all, the Nion = 9-cluster radius increases only from 32 a.u.
to 33.8 a.u. within the 37 fs pulse duration. However, the
corresponding spherical 3D cluster with (Nion − 1)3 Xe-

8

(a)

Time (a.u.)

x inner

(b)

(c)

Ninner

atom
cluster

Time (a.u.)

Space (a.u.)

ˆ = 0.062, ω = 0.057, Nion = 9, Z = 4,
FIG. 8: Result for E
(3,8,3)-pulse, and mobile ions (M = 1836). (a) Logarithmically scaled electron density, (b) xinner, and (c) Ninner (as in
Fig. 3). The blue and broken curves in (b) and (c) are the
results for immobile ions.

(a)

x inner

(b)

Time (a.u.)

(c)

Ninner

ions of charge state Z = 4 doubles its initial radius within
the same time [43]. Clearly, the Coulomb explosion is
underestimated in the 1D model because in 3D an ion
sitting at the cluster surface “sees” not only the charges
of the ions aligned along the laser field polarization direction but essentially a charged sphere containing all the
other (Nion − 1)3 − 1 ions.
Hence, in order to allow the 1D ion chain to explode
similarly to the corresponding 3D cluster one has to reduce the ion mass. To obtain the results shown in Fig. 8
M = 1836 was set. In the “worst case” introduced in
the previous paragraph the radius of the 1D cluster now
expands from 32 to ≈ 130 a.u. within 37 fs. However,
in the early stage when the wave packet formation inside the cluster takes place the charge states of the ions
are still low and the cluster radius remains close to its
original value. This is clearly visible in Fig. 8a where
between t = 300 and t = 800 the bouncing wave packet
can be easily identified. In Fig. 8b,c the dipole xinner
and the number of electrons inside the cluster Ninner are
presented, as in Fig. 3. For comparison, the results for a
computer run with immobile ions is included (blue and
broken curves). The differences are small. Only the ionization at the end of the laser pulse is slightly reduced in
the case of mobile ions. This is expected because the ions
gain their kinetic energy at the expense of the electrons.
At higher laser intensities the higher charge states are
generated earlier. In these cases the Coulomb explosion is
more violent but the wave packet mechanism contributes
little to the total ionization of the cluster anyway. In
Fig. 9 the ionization and Coulomb explosion scenario for
the highest laser intensity I = 1.14 × 1016 Wcm−2 considered in this paper is presented for the same cluster
parameters as in Fig. 8. The outer shell of all atoms is
depleted rapidly already during the first laser cycle both
for the cluster and the isolated atom. On average three
quarters of the electrons in the next shell are depopulated
during the remainder of the laser pulse in the case of the
cluster with mobile ions. The ionization probability of
the isolated atom is slightly less. The ionization of the
cluster with heavier ions is, again, higher.
Due to the rapidly increasing charge states of the
ions no formation of an electron wave packet moving
∆φ = π/2 out of phase with respect to the driving laser
field for several laser cycles can be inferred in Fig. 9a. Instead, the typical motion ∼ E(t) of free electron density
in the laser field E(t) seems to dominate the dynamics. It
is interesting to observe that the removal of the electrons
in the first shell is even more efficient for the isolated
atom than in the cluster (see Fig. 9c around t = 180).
However, shortly after “cracking” the second shell the
cluster ionization overtakes the single atom result. At
that time the xinner -plot in Fig. 9b reveals two extrema
(marked by arrows) which are approximately ∆φ = π/2
out of phase with the driving field. Thus, although a dephased wave packet dynamics over several cycles cannot
develop in such intense fields the stronger ionization of
the cluster still relies on the fact that the electron density

atom

cluster

Time (a.u.)

Space (a.u.)

ˆ = 0.57.
FIG. 9: The same as in Fig. 8 but for E

inside the cluster moves temporarily with the appropriate
phase lag necessary for efficient absorption.
IV.

CONCLUSION

The ionization dynamics of a one-dimensional rare gas
cluster model in intense and short laser pulses was investigated by means of time-dependent density functional
theory. An electron wave packet dynamics was found
to build up inside the cluster when the laser intensity
ˆ 2 was sufficiently high for modest inner ionization
I=E
but not that high that all electrons of an atomic shell are
freed within a few laser cycles. The electron wave packet
is driven into resonance with the laser field through the

9
collisions with the cluster boundary. The phase lag between the bouncing electron wave packet and the laser
field then is π/2 so that the absorption of laser energy
is particularly high. The fastest electrons in the wave
packet escape from the cluster (outer ionization). The
electric field of the bouncing electron wave packet adds
up constructively to the laser field, thus enhancing inner
ionization. This effect was called dynamical ionization
ignition. It is a robust phenomenon with respect to the
cluster size and, since it occurs during the early ionization stage, it is not affected by ionic motion. However,
with increasing laser frequency (keeping the laser intensity fixed) the mechanism is less efficient.
We expect the wave packet scenario being valid also
in real, three-dimensional rare gas clusters. Due to

the spherical cluster-vacuum boundary the wave packet
should assume a sickle-like shape in that case. However,
in order to verify this, studies with higher-dimensional
cluster models will be pursued in the future.

[1] Y. L. Shao, T. Ditmire, J. W. G. Tisch, E. Springate, J.
P. Marangos, and M. H. R. Hutchinson, Phys. Rev. Lett.
77, 3343 (1996).
[2] T. Ditmire, T. Donnelly, A. M. Rubenchik, R. W. Falcone, and M. D. Perry, Phys. Rev. A 53, 3379 (1996).
[3] T. Ditmire, J. Tisch, E. Springate, M. Mason, N. Hay,
R. Smith, J. Marangos, and M. Hutchinson, Nature 386,
54 (1997).
[4] E. Springate, N. Hay, J. W. G. Tisch, M. B. Mason, T.
Ditmire, M. H. R. Hutchinson, and J. P. Marangos, Phys.
Rev. A 61, 063201 (2000).
[5] A. McPherson, B. D. Thompson, A. B. Borisov, K. Boyer,
and C. K. Rhodes, Nature 370, 631 (1994).
[6] T. Ditmire, P. K. Patel, R. A. Smith, J. S. Wark, S. J.
Rose, D. Milathianaki, R. S. Marjoribanks, and M. H. R.
Hutchinson, J. Phys. B: At. Mol. Opt. Phys. 31, 2825
(1998).
[7] S. Ter Avetisyan, M. Schn¨
urer, H. Stiel, U. Vogt, W.
Radloff, W. Karpow, W. Sandner, and P. V. Nickles,
Phys. Rev. E 64, 036404 (2001).
[8] J. Zweiback, R. A. Smith, T. E. Cowan, G. Hays, K. B.
Wharton, V. P. Yanovsky, and T. Ditmire, Phys. Rev.
Lett. 84, 2634 (2000).
[9] K. Boyer, B. D. Thompson, A. McPherson, and C. K.
Rhodes, J. Phys. B: At. Mol. Opt. Phys. 27, 4373 (1994).
[10] W. Andreas Schroeder, T. R. Nelson, A. B. Borisov, J.
W. Longworth, K. Boyer, and C. K. Rhodes, J. Phys. B:
At. Mol. Opt. Phys. 34, 297 (2001).
[11] C. Rose-Petruck, K. J. Schafer, K. R. Wilson, and C. P.
J. Barty, Phys. Rev. A 55, 1182 (1997); C. Rose-Petruck,
K. J. Schafer, and C. P. J. Barty, Proc. SPIE Int. Soc.
Opt. Eng. 2523, 272 (1995).
[12] H. M. Milchberg, S. J. McNaught, and E. Parra, Phys.
Rev. E 64, 056402 (2001).
[13] K. Y. Kim, I. Alexeev, E. Parra, and H. M. Milchberg,
Phys. Rev. Lett. 90, 023401 (2003).
[14] Kenichi Ishikawa and Thomas Blenski, Phys. Rev. A 62,
063204 (2000).
[15] Marian Rusek, Herve Lagadec, and Thomas Blenski,
Phys. Rev. A 63, 013203 (2000).
[16] Jan Posthumus (ed.) Molecules and Clusters in Intense
Laser Fields (Cambridge University Press, Cambridge,
2001).

[17] V. P. Krainov and M. B. Smirnov, Phys. Rep. 370, 237
(2002).
[18] E. M. Snyder, S. A. Buzza, and A. W. Castleman, Jr.,
Phys. Rev. Lett. 77, 3347 (1996).
[19] Isidore Last and Joshua Jortner, Phys. Rev. A 60, 2215
(1999).
[20] Isidore Last and Joshua Jortner, Phys. Rev. A 62, 013201
(2000).
[21] Ulf Saalmann and Jan-Michael Rost, Phys. Rev. Lett.
89, 143401 (2002).
[22] Christian Siedschlag and Jan M. Rost, Phys. Rev. Lett.
89, 173401 (2002).
[23] Christian Siedschlag and Jan M. Rost, Phys. Rev. A 67,
013404 (2003).
[24] Erich Runge and E. K. U. Gross, Phys. Rev. Lett. 52,
997 (1984).
[25] Kieron Burke and E. K. U. Gross, A Guided Tour of
Time-Dependent Density Functional Theory in Density
Functionals: Theory and Applications ed. by Daniel Joubert (Springer, Berlin, 1998).
[26] E. K. U. Gross, J. F. Dobson, and M. Petersilka, in Topics
in Current Chemistry (Springer, Berlin, 1996) pp. 81.
[27] F. Calvayrac, P.-G. Reinhard, E. Suraud, and C. A. Ullrich, Phys. Rep. 337, 493 (2000).
[28] D. Bauer, F. Ceccherini, A. Macchi, and F. Cornolti,
Phys. Rev. A 64, 063203 (2001).
[29] Miroslav Brewczyk, Charles W. Clark, Maciej Lewenstein, and Kazimierz Rz¸az˙ ewski, Phys. Rev. Lett. 80,
1857 (1998).
[30] Miroslav Brewczyk and Kazimierz Rz¸az˙ ewski, Phys. Rev.
A 60, 2285 (1999).
[31] Val´erie V´eniard, Richard Ta¨ieb, and Alfred Maquet,
Phys. Rev. A 65, 013202 (2001).
[32] I. Grigorenko, K. H. Bennemann, and M. E. Garcia, Europhys. Lett. 57, 39 (2002).
[33] The dipole approximation can be applied because the
wavelengths of the laser light under consideration are
many times greater than the cluster diameters in our
study, and magnetic field effects do not play a role.
[34] J. B. Krieger, Yan Li, and G. J. Iafrate, Phys. Rev. A
45, 101 (1992).
[35] F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and
G. Zwicknagel, J. Phys. B: At. Mol. Opt. Phys. 36, 273

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (D.B.) and through the INFM advanced
research project Clusters (A.M.). The permission to run
our codes on the Linux cluster at PC2 in Paderborn, Germany, is gratefully acknowledged.

10
(2003).
[36] On the other hand, experimental results could several
times successfully be interpreted by employing the NP
model where collisional absorption is supposed to be essential [4, 37]. Whether the success of the NP model in
this context is just by coincidence or collisional absorption is really important is not yet clear.
[37] M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, J.
Phys. B: At. Mol. Opt. Phys. 30, L251 (1997).
[38] F. Brunel, Phys. Rev. Lett. 59, 52 (1987); F. Brunel,
Phys. Fluids 31, 2714 (1988).
[39] P. Gibbon and A. R. Bell, Phys. Rev. Lett. 68, 1535
(1992).
[40] P. Mulser, H. Ruhl, and J. Steinmetz, Las. Part. Beams
19, 23 (2001).
[41] Note, however, that xinner(t) is the dipole and does

not describe the center of the oscillating wave packet.
The center of mass of all inner electrons is xc.m. =
xinner /Ninner . However, to xinner, xc.m. , and Ninner also
the electrons that are still bound to their parent ions contribute so that xc.m. ≪ R holds despite the fact that the
oscillating wave packet has an excursion ≈ R.
[42] H. Wabnitz, L. Bittner, A. R. B. de Castro, R.
D¨
ohrmann, P. G¨
urtler, T. Laarmann, W. Laasch, J.
Schulz, A. Swiderski, K. van Haeften, T. M¨
oller, B. Faatz,
A. Fateev, J. Feldhaus, C. Gerth, U. Hahn, E. Saldin, E.
Schneidmiller, K. Sytchev, K. Tiedke, R. Treusch, and
M. Yurkov, Nature 420, 482 (2002).
[43] The time during which the cluster radius doubles reads
td = 1.6 [M R3 /(Nion − 1)3 Z 2 ]1/2 (cf. Eq. (42) in [17]).